Volume Formulas

The volume formula is a mathematical expression used to find the total space (vacuum) occupied by any three-dimensional object. Let us understand in detail about volume formulas of different 3-D shapes.

What is Volume Formula?

The formula used to calculate the total cubic capacity that an object can hold is its volume formula. The unit of volume of a 3-d shape is expressed as units³ or cubic units. Look at the volume formulas chart below depicting the volume formulas of respective 3-D shapes.

Let us learn about the general volume formulas of various shapes in detail.

Volume Formulas of 3-D Shapes

We now know that the volume formula is used to calculate the volume of a three-dimensional object. In this section, we will learn about the volume formulas with respective dimensions of different 3-D shapes. 

Volume Formula of Cube

The volume formula of a cube depends upon the three sides of a cube where all three sides are equal in measure. The volume of a cube is the quantity occupied by the cube. The general formula of the volume of a cube is given as:

• The volume of a cube = a × a × a = a³ cubic units, where 'a' is the length of the side of the cube.
• The volume of a cube formula using diagonal can be given as V = (√3×d³)/9, where d is the length of the diagonal of the cube.

Volume Formula of Cuboid

To calculate the amount of space enclosed by the cuboid, we use the volume of a cuboid formula. The general formula of the volume of a cuboid is mathematically expressed as:

• The volume of cuboid = Base Area × Height cubic units
• The base area for cuboid = l × b square units
• Hence, the volume of a cuboid, V = l × b × h = lbh units³, where 'l' 'b' and 'h' represent the length, breadth, and height of the cuboid.

Volume Formula of Cone

To calculate the amount of space occupied within a 3-D shape cone that has a circular base with a radius 'r' and height 'h' we use the volume formula of a cone. The general volume formula of the cone is expressed as:

The volume of a cone, V = (1/3)πr²h cubic units.

Here,

• 'r' is the radius of the base (circle) of the cone
• 'h' is the height of the cone
• π is a constant with the value either 22/7 (or) 3.142.

Volume Formula of Cylinder

The volume formula of the cylinder is used to determine the amount of space (capacity) occupied inside it. We know that the base of a right circular cylinder is a circle and the area of a circle of radius 'r' is πr². Thus, the volume of a cylinder formula is,

The volume of a cylinder = πr²h cubic units

Here,

• 'r' is the radius of the base (circle) of the cylinder
• 'h' is the height of the cylinder
• π is a constant whose value is either 22/7 (or) 3.142.

Thus, the volume of a cylinder directly proportional to its height and to the square of its radius. i.e., the volume of the cylinder becomes four times, if the radius of the cylinder is double. 

Volume Formula of Sphere

A football is a perfect example that resembles a shape of a sphere. It is a three-dimensional solid object with a round structure. The amount of air that is filled in a ball is termed as the volume of a sphere or a ball. The volume formula of the sphere is given as:

The volume of sphere = (2/3)πr²h
If the diameter of the sphere = 2r
Hence, volume of sphere is (2/3)πr²h = (2/3)πr²(2r) = (4/3)πr³ cubic units

The volume of a sphere is (4/3)πr³ cubic units

Here,

• 'r' is the radius of the sphere
• 'h' is the height of the sphere
• π is a constant whose value is either 3.142 or 22/7.

Volume Formula of Hemisphere

Hemisphere is half of a sphere, we can easily derive the volume formula of the hemisphere by using the volume formula of the sphere. Now considering that the radius of a sphere is 'r' units and the volume of the sphere is (4/3)πr³.

Thus, the volume of hemisphere can be given as:V = ½ (4/3)πr³

Volume of hemisphere = (2/3)πr³ cubic units

Here,

• 'r' is the radius of the hemisphere
• π is a constant whose value is either 3.142 or 22/7.

Volume Formula of Prism

The volume formula of a prism is given by the product of the area of the base and height of the prism. It is mathematically expressed as:

The volume of prism V =  B × h units³.

Here,

• "B" is the base area in square units
• "h" is the height of the prism in units.

There are seven types of prisms based on the shape of the bases of prisms. The volume formula of prisms depends on the different bases of the prisms. Check out the volume of prism to understand the concept behind the volume formulas of various prisms.

Volume Formula of Pyramid

The volume of a pyramid is one-third of the volume of the prism (i.e., their bases and heights are congruent). Thus,

The volume of pyramid(V) = (1/3) (Bh) units³, where

• B = base area of the pyramid in square units
• h = Height of the pyramid (altitude) in units

Examples on Volume Formula 

Example 1: A cylindrical tank has a radius of 3 units and a height of 8 units, using the volume formula find the volume of the cylinder find its surface area.

Solution:

Given: r = 3 units, h = 8 units
On substituting the values in the volume formula of the cylinder we have,
Volume of a cylinder = πr²h
V = π(3)²(8)
V = π × 9 × 8
V = 72 π
Substituting the value of π = 3.14
V =  72 × 3.14 = 226.08 units³
Volume of cylinder is 226.08 units³

Example 2: Given that the radius of a cone is 4 units and the height of a cone is 9 units. Using the volume formula, determine the volume of a cone.

Solution:

Given: Radius = 4 units and height =9 units
Volume formula of cone = (1/3)πr²h.
=1/3 × 3.14 × 4² × 9
=1/3 × 452.16
=150.72 units³
∴The volume of the cone will be 150.72 units³

Example 3: Using the volume formula of the cube find the volume of the cuboid whose length is 9 inches, breadth is 7 inches and height is 5 inches.

Solution: Given length of cuboid = 9 inches, breadth of cuboid = 7 inches, and height of cuboid = 5 inches.
The volume formula of cuboid = l × b × h
On substituting the values of l, b, and h in the volume formula we have,
V = 9 × 7 × 5
= 315
= 315 inches³
∴The volume of a cuboid will be 315 inches²